zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science. (English) Zbl 1231.35288
Summary: We suggest a fractional functional for the variational iteration method to solve the linear and nonlinear fractional order partial differential equations with fractional order initial and boundary conditions by using the modified Riemann-Liouville fractional derivative proposed by G. Jumarie [Appl. Math. Lett. 22, No. 3, 378–385 (2009; Zbl 1171.26305)]. A fractional order Lagrange multiplier is considered. The solution is plotted for different values of α.
35R11Fractional partial differential equations
35K15Second order parabolic equations, initial value problems
35L15Second order hyperbolic equations, initial value problems
35L20Second order hyperbolic equations, boundary value problems
45K05Integro-partial differential equations
65M99Numerical methods for IVP of PDE
[1]Binning, P.; Celia, M. A.: Practical implementation of the fractional flow approach to multi-phase flow simulation, Advan. watr. Resour. 22, 461-478 (1999)
[2]Shen, C.; Phanikumar, M. S.: An efficient space-fractional dispersion approximation for stream solute transport modeling, Advan. watr. Resour. 32, 1482-1494 (2009)
[3]Huang, Q.; Huang, G.; Zhan, H.: A finite element solution for the fractional advection–dispersion equation, Advan. watr. Resour. 31, 1578-1589 (2008)
[4]Wheatcraft, S. W.; Meerschaert, M. M.: Fractional conservation of mass, Advan. watr. Resour. 31, 1377-1381 (2008)
[5]Dozier, J.; Painter, T. H.; Rittger, K.; Frew, J. E.: Time-space continuity of daily maps of fractional snow cover and albedo from MODIS, Advan. watr. Resour. 31, 1515-1526 (2008)
[6]Kevorkian, J.; Cole, J. D.: Multiple scale and singular perturbation method, (1996)
[7]He, J. H.: Homotopy perturbation technique, Comput. math. Appl. mech. Engy., 178-257 (1999)
[8]Yildirim, A.; Kocak, H.: Homotopy perturbation method for solving the space–time fractional advection–dispersion equation, Advan. watr. Resour. 32, 1711-1716 (2009)
[9]Ganji, D. D.; Ganji, S. S.; Karimpour, S.; Ganji, Z. Z.: Numerical study of homotopy-perturbation method applied to Burgers equation in fluid, Numer. methods partial differential equations 26, 917-930 (2010)
[10]Khan, Y.; Wu, Q.: Homotopy perturbation transform method for nonlinear equations using he’s polynomials, Comput. math. Appl. 61, 1963-1967 (2011) · Zbl 1219.65119 · doi:10.1016/j.camwa.2010.08.022
[11]Nadeem, S.; Akbar, N. S.: Peristaltic flow of a Jeffrey fluid with variable viscosity in an asymmetric channel, Z. naturforsch. 64a, 713-722 (2009)
[12]Nadeem, S.; Akbar, N. S.: Influence of heat transfer on a peristaltic transport of Herschel Bulkley fluid in a non-uniform inclined tube, Commun. nonlinear sci. Numer. simul. 14, 4100-4113 (2009) · Zbl 1221.76269 · doi:10.1016/j.cnsns.2009.02.032
[13]Nadeem, S.; Akbar, N. S.: Influence of heat transfer on a peristaltic flow of Johnson Segalman fluid in a non uniform tube, International communications in heat and mass transfer 36, 1050-1059 (2009)
[14]Nadeem, S.; Hayat, T.; Akbar, Noreen Sher; Malik, M. Y.: On the influence of heat transfer in peristalsis with variable viscosity, International journal of heat and mass transfer 52, 4722-4730 (2009) · Zbl 1176.80030 · doi:10.1016/j.ijheatmasstransfer.2009.04.037
[15]He, J. H.: Variational iteration method–a kind of non-linear analytical technique: some examples, Int. J. Non-linear mech. 34, 699-708 (1999)
[16]He, J. H.; Wu, G. C.; Austin, F.: The variational iteration method which should be followed, Nonl. sci. Lett. A 1, 1-30 (2010)
[17]Faraz, N.; Khan, Y.; Austin, F.: An alternative approach to differential-difference equations using the variational iteration method, Z. naturforsch. 65a, 1055-1059 (2010)
[18]Al-Khaled, K.; Momani, S.: An approximate solution for a fractional diffusion-wave equation using the decomposition method, Appl. math. Comput. 165, 473-483 (2005) · Zbl 1071.65135 · doi:10.1016/j.amc.2004.06.026
[19]Khan, Y.; Faraz, N.: Modified fractional decomposition method having integral (dξ)α, J. King saud. Uni. sci. 23, 157-161 (2011)
[20]Khan, Y.: An effective modification of the Laplace decomposition method for nonlinear equations, Int. J. Nonlinear sci. Numer. simul. 10, 1373-1376 (2009)
[21]Khan, Y.; Faraz, N.: Application of modified Laplace decomposition method for solving boundary layer equation, J. King saud. Uni. sci. 23, 115-119 (2011)
[22]Khan, Y.; Austin, F.: Application of the Laplace decomposition method to nonlinear homogeneous and non-homogenous advection equations, Z. naturforsch. 65a, 849-853 (2010)
[23]Nadeem, S.; Akbar, N. S.: Effects of heat transfer on the peristaltic transport of MHD Newtonian fluid with variable viscosity: application of Adomian decomposition method, Commun. nonlinear sci. Numer. simul. 14, 3844-3855 (2009)
[24]He, J. H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. methods appl. Mech. eng. 167, 57-68 (1998) · Zbl 0942.76077 · doi:10.1016/S0045-7825(98)00108-X
[25]Odibat, Z.; Momani, S.: The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Comput. math. Appl. 58, 2199-2208 (2009) · Zbl 1189.65254 · doi:10.1016/j.camwa.2009.03.009
[26]Das, S.: Analytical solution of a fractional diffusion equation by variational iteration method, Comput. math. Appl. 57, 483-487 (2009) · Zbl 1165.35398 · doi:10.1016/j.camwa.2008.09.045
[27]Momani, S.; Odibat, Z.: Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Comput. math. Appl. 58, 2199-2208 (2009)
[28]Momani, S.; Odibat, Z.: Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. lett. A 355, 271-279 (2006)
[29]Momani, S.; Odibat, Z.: Numerical comparison of the methods for solving linear differential equations of fractional order, Chaos solitons fractals 31, 1248-1255 (2007) · Zbl 1137.65450 · doi:10.1016/j.chaos.2005.10.068
[30]Faraz, N.; Khan, Y.; Yildirim, A.: Analytical approach to two-dimensional viscous flow with a shrinking sheet via variational iteration algorithm-II, J. King saud. Uni. sci. 23, 77-81 (2011)
[31]Inc, Mustafa: The approximate and exact solutions of the space and time-fractional Burgers equations with initial conditions by variational iteration method, J. math. Anal. appl. 345, 476-484 (2008) · Zbl 1146.35304 · doi:10.1016/j.jmaa.2008.04.007
[32]Inokuti, M.; Sekine, H.; Mura, T.: General use of the Lagrange multiplier in nonlinear mathematical physics, Variational methods in the mechanics of solids, 156-162 (1978)
[33]Jumarie, G.: Table of some basic fractional calculus formulae derived from a modified Riemann–Liouville derivative for non-differentiable functions, Appl. math. Lett. 22, 378-385 (2009) · Zbl 1171.26305 · doi:10.1016/j.aml.2008.06.003
[34]Podlubry, I.: Fractional differential equations, (1999)
[35]Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations, J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194
[36]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (2003)
[37]Jumarie, G.: New stochastic fractional models for malthusian growth, the Poissonian birth process and optimal management of populations, Math. comput. Model. 44, 231-254 (2006) · Zbl 1130.92043 · doi:10.1016/j.mcm.2005.10.003
[38]Jumarie, G.: Laplace’s transform of fractional order via the mittage–Leffler funcation and modified Riemann–Liouville derivative, Appl. math. Lett. 22, 1659-1664 (2009) · Zbl 1181.44001 · doi:10.1016/j.aml.2009.05.011
[39]Wu, G. C.; He, J. H.: Fractional calculus of variations in fractal sapcetime, Nonlinear sci. Lett. A 1, No. 3, 281-287 (2010)
[40]Wu, G. C.; Lee, E. W. M.: Fractional variational iteration method and its application, Phys. lett. A 374, 2506-2509 (2010)